Problem 70 - Algebra A Combined Function

Question

Perform each indicated operation. Write all results in lowest terms. $$ \frac{2}{3}+\frac{3}{7} $$

Step-by-Step Solution

Answer

\( \frac{23}{21} \) is the sum.

Step 1: Understand the Problem

We need to add two fractions: \( \frac{2}{3} \) and \( \frac{3}{7} \). To do this, the fractions must have a common denominator.

Step 2: Find the Least Common Denominator

The denominators of the given fractions are 3 and 7. The least common multiple (LCM) of 3 and 7 is 21, which will be used as the common denominator.

Step 3: Convert Each Fraction to the Common Denominator

Convert \( \frac{2}{3} \) to a fraction with a denominator of 21: \( \frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21} \). Convert \( \frac{3}{7} \) to a fraction with a denominator of 21: \( \frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21} \).

Step 4: Add the Fractions

Now add the two fractions, \( \frac{14}{21} + \frac{9}{21} \), by adding their numerators:\[ \frac{14 + 9}{21} = \frac{23}{21} \].

Step 5: Simplify the Result if Possible

The fraction \( \frac{23}{21} \) is already in simplest form because 23 is a prime number and does not divide evenly into 21. Hence, \( \frac{23}{21} \) is the answer.

Key Concepts

Least Common Denominator

When adding fractions like \( \frac{2}{3} \) and \( \frac{3}{7} \), the first thing we need to do is find the Least Common Denominator (LCD). This is because fractions must have the same denominator to be added together. In simple terms, the LCD is the smallest number that both denominators can divide into without leaving a remainder.

For the fractions \( \frac{2}{3} \) and \( \frac{3}{7} \), the denominators are 3 and 7. To find the LCD, we identify the least common multiple (LCM) of these two numbers. The LCM of 3 and 7 is 21, which will be our common denominator.

Finding the LCD is crucial because it allows us to rewrite each fraction with a common denominator, enabling us to perform the addition. It's like finding a common language that both fractions can "speak."

Simplifying Fractions

Once we've added the fractions and have a result like \( \frac{23}{21} \), we need to simplify the fraction if possible. Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1.

To check if a fraction can be simplified, find the greatest common divisor (GCD) of the numerator and the denominator. Divide both by the GCD to see if you can simplify the fraction further. In our example, \( 23 \) and \( 21 \) don't share any common factors other than 1, indicating that \( \frac{23}{21} \) is already in its simplest form.

• Look for common factors
• Check divisibility
• Use the GCD to simplify if needed

Simplifying fractions is like cleaning up to get the neatest version.

Prime Numbers

Prime numbers play a key role in simplifying fractions and understanding why certain fractions cannot be simplified further. A prime number is one that has only two distinct positive divisors: 1 and itself.

In our fraction \( \frac{23}{21} \), 23 is a prime number. This indicates it cannot be divided evenly by any number other than 1 and 23 itself. The number 21, on the other hand, is not prime because it is divisible by 1, 3, 7, and 21.

Understanding prime numbers helps us recognize when a fraction is fully simplified. In the case where the numerator or denominator is prime, like 23 in this exercise, and shares no common factors with the other non-prime denominator, it signals that we have reached the simplest form. This is crucial for fraction simplification and for recognizing numbers that can’t be broken down any further.